Rounding numbers to a given accuracy
Number • Measures and accuracy
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Key concepts
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Definition of rounding to a given accuracy
Rounding to a given accuracy means replacing a number by another number that is simpler but close in value, specified by decimal places or significant figures. The specified accuracy defines the digit that remains; digits after that position determine whether the remaining digit increases or stays the same. Limiting factors include the required place or count of significant figures and the context of measurement. Measurements with limited precision produce larger rounding intervals and wider upper and lower bounds.
Rounding to decimal places
Rounding to a given number of decimal places keeps that many digits after the decimal point and removes digits further right. The digit immediately to the right of the required decimal place controls the change: if that digit is 0–4, the required digit stays the same; if it is 5–9, the required digit increases by one and remaining digits become zero or are dropped. Cause: the digit to the right determines a midpoint that splits values. Effect: numbers on or above the midpoint round up, and numbers below the midpoint round down, producing a single rounded value for all numbers in that interval.
Rounding to significant figures
Rounding to significant figures preserves a fixed count of meaningful digits starting from the first non-zero digit. The digit immediately after the last required significant figure controls whether the last significant digit increases or stays the same. Limiting factors include leading zeros, which do not count as significant, and the overall magnitude: the same number of significant figures gives coarser precision for large numbers and finer relative precision for small numbers.
Rounding to nearest 10, 100, 1000, etc.
Rounding to the nearest 10, 100, 1000, or other power of ten removes the smaller place values and sets them to zero. The digit immediately to the right of the chosen place controls the change: 0–4 causes rounding down, 5–9 causes rounding up of the chosen place. Cause: grouping values by intervals equal to the rounding unit. Effect: all numbers in the same interval share the same rounded result, and the absolute error can be up to half the rounding unit.
Upper and lower bounds
Upper and lower bounds describe the range of original values that round to a given rounded value. The lower bound equals the rounded value minus half the unit of accuracy (exclusive or inclusive depending on convention); the upper bound equals the rounded value plus half the unit of accuracy (exclusive or inclusive depending on convention). Limiting factors include the rounding convention for exact midpoints. For most contexts, the interval is half-open: the lower bound is inclusive and the upper bound is exclusive, which avoids overlap between adjacent rounded values.
Common pitfalls and best practice
Rounding too early in a multi-step calculation causes cumulative error; therefore retain extra significant figures through intermediate steps and round at the end unless instructed otherwise. Misidentifying significant figures (counting leading zeros) causes incorrect rounding. Cause: premature rounding reduces precision available for later operations. Effect: final answers deviate more from true values, especially in subtraction or division where relative error magnifies.
Key notes
Important points to keep in mind